Feb 18, 2022

Christian Gourieroux, Joann Jasiak

# 1 Introduction

The macroeconomic and financial models used in practice provide reliable predictions at short horizons of 1 to 5 years. Recently, there has been a growing interest in providing long run predictions at horizons of 10 to 50 years, in the context of transition to low carbon economy, climate risk and rare extreme events, for evaluating of necessary behavioral and technical changes. In the financial sector as well, the long run predictions may soon become mandatory for prudential supervision in banks and insurance companies.

The long-run predictions are difficult to compute for several reasons. The standard prediction models with short lags and the associated statistical inference methods are inadequate for long horizons. Moreover, long run predictions at horizons of 50 or 100 years are difficult to compute from macroeconomic or financial time series observed over periods shorter or equal to the prediction horizon of interest. Therefore, the long run predictions remain mainly model based. As such, they can be improved so that:

i) The estimation methods account for the long run component, even though it is difficult to detect over the sampling period.

ii) The long run predictions produce reasonable outcomes in the sense that point forecasts should take values from the set of admissible values of the predicted variable.

iii) The prediction errors are not underestimated due to the selected dynamic model and its estimation method.

This paper examines the feasibility of such improvements in the class of dynamic stochastic linear models with long run properties already known in the literature to some extent. The Structural Vector Autoregressive (SVAR) models are stochastic linear systems which are second-order identifiable under a set of parameter restrictions. The SVAR models are used in macroeconomics, monetary economics and macro-finance for the analysis of multivariate economic processes and prediction of future shock effects through impulse response functions.

## 1.1 SVAR Models and their Limitations

Two types of SVAR models can be distinguished, which are the stationary (regular) SVAR models and cointegrated SVAR models with nonstationary unit root components. While both types of SVAR models are efficient instruments of short run analysis, they have limitations in application to the long run and ultra long run analysis.

i) The presence of nonstationary features, such as nonstationary unit roots, implies explosive patterns in trajectories, which are incompatible with the behavior of variables such as the growth rate of per capita real GDP, productivity, real food expenditure per capita, interest rates, real exchange rates, and some commodity prices, or spot-forward spreads.

ii) The identification restrictions imposed on the long run behaviour, i.e. the so-called long run identification restrictions can affect the long-run predictions and the long run patterns of impulse response functions as well as their signs.

iii) The long run risk is diversified away and often disregarded in stationary SVAR models.

## 1.2 An Alternative Approach to SVAR Models

Among the macroeconomic time series, those considered stationary often display persistence up to high lags, in the sample autocorrelation functions. Therefore, a large body of literature use the local-to-unity model linking the aforementioned two types of SVAR dynamics and develop associated inference methods. However, the standard local-to-unity models may not be suitable for stationary time series (see the discussion in Appendix 1).

The aim of our paper is to propose an alternative approach by considering a stationary VAR model with a stationary short run (SR) component and a stationary multivariate ultra long run (ULR) component with asymptotic unit root and close to zero sigma,

For illustration, let us consider the framework of stationary Gaussian processes (where strict stationarity and second-order stationarity are equivalent). This simplified framework is not only convenient for comparing our analysis with the large body of literature on structural SVAR models, but also for introducing and interpreting the notion of an ultra long run process in a stationary time series seen as a sum of a regular and a singular components. There exist two generic representation theorems of a time series seen the Wold representation theorem for weakly stationary processes and the Volterra representation theorem for strictly stationary processes. These two representations coincide for the Gaussian stationary processes.

## 1.3 The Framework of the Study

According to the Wold representation, any (multivariate) stationary Gaussian process y(t) can be written in such a way that the moving average component is the regular component. The component Zt is the singular component of process (y(t)) and is measurable with respect to Ft. Following Doob (1944), this singular component is also referred to the deterministic component of the process. This explains why in practice, Zt is often disregarded and replaced by a deterministic constant.

The term "deterministic" is misleading, because the stationary singular component can be constant over time while being stochastic. This fact that Zt is measurable with respect to Ft means that it is only influenced by the infinitely distant past and has a long run interpretation. In particular, the singular component is a stationary martingale, which is a particular stationary unit root process. The singular component can be approximated by a moving average and therefore has a significant impact in the long run dynamic and is therefore interpreted as the Ultra Long Run (ULR) component. Moreover, process fyT(t)g remains stationary, but becomes non-ergodic.

More generally, in the VAR framework we can write the n-dimensional process yT(t) as a sum of the short run (SR) component ys(t), which for ease of exposition is assumed to follow a VAR(1) process, and an ULR component as described in our proposed formula. The component [yl(ζ)] follows a L-dimensional (continuous time) multivariate Ornstein-Uhlenbeck process.

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